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Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2011, Article ID 743916, 13 pages doi:10.1155/2011/743916 Research Article Multimode Transmission in Network MIMO Downlink with Incomplete CSI Nima Seifi,1 Mats Viberg (EURASIP Member),1 Robert W Heath Jr.,2 Jun Zhang,3 and Mikael Coldrey4 Department of Signals and Systems, Chalmers University of Technology, 412 96 Gothenburg, Sweden Wireless Networking and Communications Group (WNCG), Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712-0240, USA Deptartment of Electronic and Computer Engineering, Hong Kong University of Science and Technology, (HKUST), Clear Water Bay, Kowloon, Hong Kong Ericsson Research, Ericsson AB, 417 56 Gothenburg, Sweden Correspondence should be addressed to Nima Seifi, nima.seifi@chalmers.se Received June 2010; Accepted 16 October 2010 Academic Editor: Francesco Verde Copyright © 2011 Nima Seifi et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We consider a cooperative multicell MIMO (a.k.a network MIMO) downlink system with multiantenna base stations (BSs), which are connected to a central unit and communicate with multiantenna users In such a network, obtaining perfect channel state information (CSI) of all users at the central unit to exploit opportunistic scheduling requires a substantial amount of feedback and backhaul signaling We propose a scheduling algorithm based only on the knowledge of the average SNR at each user from all the cooperating BSs, denoted as incomplete CSI Multimode transmission is applied that is able to adaptively adjust the number of data streams transmitted to each user Utilizing the results of random matrix theory, an analytical framework is proposed to approximate the ergodic rate of each user with different number of data streams Using these ergodic rates, a joint user and mode selection algorithm is proposed, where only the scheduled users need to feed back instantaneous CSI Simulation results demonstrate that the developed analytical framework provides a good approximation for a practical number of antennas While substantially reducing the feedback overhead, it is shown that the proposed scheduling algorithm performs closely to the opportunistic scheduling algorithm that requires instantaneous CSI feedback from all users Introduction Recently, cooperative multicell transmission (also called network MIMO) has been proposed as an efficient way to suppress the intercell interference and increase the downlink capacity of cellular systems [1–3] In one way of realizing a network MIMO system, multiple base stations (BSs) are connected to a central unit via backhaul links The central unit coordinates BSs and performs joint scheduling and signal processing operations Assuming no limitations with regards to the capacity, error, and delay in the backhaul, and upon the availability of perfect channel state information (CSI) of all users at the central unit, the network MIMO system in the downlink is equivalent to a MIMO broadcast channel with per-BS power constraint (PBPC) [4–7] Many of the previous studies on network MIMO assume full coordination over the whole system, which is not practical (if not impossible) First, the backhaul links connecting the BSs and the central unit are subject to transmission error [8] and delay [9] They also have limited capacity which confines the amount of data and CSI sharing [10– 12] Second, connecting a large number of BSs for joint processing is of high complexity, which has motivated the development of coordination strategies at a local scale [13, 14] Third, obtaining perfect CSI from all the users at the central unit, which is indispensable to achieve the full diversity or multiplexing gains, results in a substantial training and feedback overhead [15–17] In this paper, we consider local BS cooperation and focus on the third limitation, that is, the substantial overhead to obtain the CSI for each active user at the central unit To this end, we assume the backhaul links to be perfect and leave the study of the effect of imperfect backhaul to future work We propose a framework that enables scheduling based only on the knowledge of the average received SNR at each user from all the cooperating BSs, denoted as incomplete CSI This reduces the overhead both on the feedback channel and the backhaul, since only the selected users (usually a small number) have to feedback their instantaneous CSI for precoder design 1.1 Related Work The scheduling problem in the singlecell multiuser MIMO downlink has been widely investigated under various precoding and beamforming strategies [18– 22] The total number of transmitted data streams in such systems is upper bounded by the number of BS antennas under a linear precoding framework Therefore, if the total number of receive antennas in the system is greater than the total number of transmit antennas, the scheduling will consist of selecting both users and the number of data streams or modes (note that the term “mode” used in this paper denotes the number of data streams for a given user rather than the number of active users, as in [22] or different MIMO transmission techniques, such as spatial multiplexing/diversity mode [23]) to each user Such multimode transmission improves performance by allowing a dynamic allocation of the transmission resources among the users [24] User and mode selection in a network MIMO system is more challenging than in its single-cell counterpart The increased number of users and BSs in the network MIMO makes the CSI requirement daunting Acquiring CSI at the central unit is one of the limiting factors for a practical network MIMO system The availability of perfect CSI, however, has a cardinal role in exploiting the spatial degrees of freedom in such systems [13] In practice, CSI for the downlink is obtained through some form of training and feedback In time-division duplexing (TDD) systems, the CSI is obtained at each BS using the channel reciprocity (see, e.g., [25]) In frequency-division duplexing (FDD) systems, since uplink and downlink take place in widely separated frequency bands, the downlink CSI is fed back via some explicit feedback channels (see e.g., [26]) This places a significant burden on the uplink feedback channel The feedback overhead increases with the number of BSs, users, antennas, and subcarriers and can easily occupy the whole uplink resources Furthermore, in both TDD and FDD systems, the CSI should be forwarded from the BSs to the central unit which limits the backhaul resources for data transmission The tradeoff between the resources dedicated to CSI overhead and data transmission in the backhaul has been recently studied in [15–17], where several multicell system architectures were compared It was further shown that the downlink performance of network MIMO systems is mainly limited by the inevitable acquisition of CSI rather than by limited backhaul capacity As a solution to this limitation, some authors [27, 28] have proposed strategies based on local CSI at the BSs EURASIP Journal on Advances in Signal Processing and statistical CSI at the central unit, whereas others [29, 30] consider to serve only certain subsets of users with multiple BSs In [31], a decentralized cooperation framework has been proposed in which all the necessary processing is performed in a truly distributed manner among the BSs without the need of any CSI exchange with the central unit Several BS cooperation strategies have been studied which consider the combination of limited-capacity backhaul and imperfect CSI [32, 33] 1.2 Contributions In this paper, we develop a scheduling algorithm for a network MIMO system with multiantenna users To reduce the feedback overhead, we adopt a twostep scheduling process: the first step is joint user and mode selection, and the second step is the feedback and precoder design which only involves the users selected in the first step The two-step multimode transmission strategy was also proposed in the single-cell MIMO downlink in [22], with single-antenna users and imperfect CSI at the BS The main contributions are as follows Ergodic Rate Analysis We propose an analytical framework to compute an accurate approximation for the ergodic rate of each user with different number of data streams, based only on the knowledge of incomplete CSI for any given location Essentially, the aggregate channel from multiple distributed cooperating BSs can be well approximated as coming from a single super BS This enables an efficient method to evaluate the performance of network MIMO systems without the need for extensive and computationally intensive MonteCarlo simulations Joint User and Mode Selection Algorithm We use the derived ergodic user rates as a metric to perform user and mode selection, which is suitable for the data application without stringent delay constraint Since the ergodic rate for each user is obtained only based on incomplete CSI, the small-scale fading is not exploited in the proposed strategy Therefore, it does not provide small-scale multiuser diversity gain, but instead, it omits the need for feedback of instantaneous CSI from a large number of users for scheduling by more than 93% It is also shown that the performance of the proposed user and mode-selection strategy is very close to the opportunistic scheduling based on instantaneous CSI feedback from all the users 1.3 Organization The rest of the paper is organized as follows: the system model and transmission strategy are described in Section In Section 3, some mathematical preliminaries, which are useful throughout the paper, are presented An analytical framework to derive an approximation for the ergodic rate of each user at different modes is proposed in Section A greedy joint user and mode selection algorithm based on the derived ergodic rates of each user is described in Section The performance of the proposed user and mode-selection algorithm is evaluated in Section Finally, Section concludes the paper and discusses the future work EURASIP Journal on Advances in Signal Processing 1.4 Notation Scalars are denoted by lowercase letters, vectors denoted by boldface lowercase letters, and matrices denoted by boldface uppercase letters (·) , (·)∗ , det(·), log(·), and tr(·) denote transpose, complex conjugate transpose, determinant, 2-base logarithm, and trace of a matrix, respectively; U(m, n) is the collection of m × n unitary matrices with unit-norm orthogonal columns E[·] is the statistical expectation [Φ]:(m:n) denotes the matrix obtained by choosing n − m + columns from Φ starting from the mth column [Φ](m,n) denotes the m × n upper-left corner of a square matrix Φ λi (Φ) and λmin (Φ) denote the ith ordered and the smallest eigenvalue of ΦΦ∗ , respectively x denotes the Euclidean norm of a complex vector x, and |S | is the cardinality of a set S; dim(·) is the dimensionality operator Further, · denotes the floor operation, Φ ⊗ Ψ denotes the Kronecker product of the two matrices Φ and n Ψ, and Cm denotes the combination of n choosing m System Model 2.1 Network MIMO Structure The network MIMO system considered in this paper comprises B cells, each of which has a BS with Nt antennas and Kb users, each equipped with Nr antennas, for Kb ≥ and b = 1, 2, , B The total number of active users in the system is denoted as K = B b=1 Kb Users in different locations of the cellular coverage are subject to distance-dependent pathloss and shadowing A narrowband frequency-flat fading channel is considered We consider the downlink transmission The following are the key assumptions made in this paper Assumption All the B cooperating BSs are interconnected via a central unit with the use of backhaul links with infinite capacity such that they can fully share CSI and user data With this assumption, all the cooperating BSs form a distributed antenna array that can perform joint scheduling and transmission Assumption The number of antennas at each BS is greater than that of each user, that is, Nt ≥ Nr Due to space constraints, user terminals can only have a small number of antennas, which makes Nt ≥ Nr a reasonable assumption Therefore, each user can have at most Nr data streams Assumption For the scheduling phase, the central unit relies on the knowledge of incomplete CSI of all users (scheduling CSI) For the transmission phase, however, the central unit has perfect knowledge of the singular values and the corresponding right singular vectors of the selected eigenmodes of each selected user (transmission CSI) for precoding design This assumption of scheduling CSI significantly reduces the feedback and backhaul signaling overhead for scheduling The transmission CSI assumption is due to the transmission strategy employed in this paper (see Section 2.3), which reduces the transmission CSI required for precoding design with respect to the strategies which requires the complete channel matrix of each selected user The transmission CSI can be reduced even more using limited feedback techniques [34–36], which we will not explore in this paper 2.2 Received Signal Model The aggregate channel matrix of user k from all the B cooperating BSs can be written as Hk = √ ρk,1 Hk,1 · · · √ ρk,B Hk,B , (1) where Hk,b ∈ CNr ×Nt represents the small-scale fading channel matrix and ρk,b is the large-scale fading channel coefficient that captures the distance-dependent pathloss including shadowing for user k from the bth BS We denote the Nt × transmit signal vector from the bth BS as xb Therefore, the BNt × aggregate transmit signal vector from all the B cooperating BSs can be written as x = x1 · · · xB (2) The discrete-time complex baseband signal received by the kth user is given by yk = H k x + n k , (3) where nk is the noise vector at the kth user, with entries that are independent and identically distributed complex Gaussian with zero mean and unit variance, denoted as i.i.d CN (0, 1) 2.3 Transmission Strategy To simultaneously transmit multiple spatially multiplexed streams to multiple users, we adopt a linear precoding strategy called multiuser eigenmode transmission (MET) (the framework developed in this paper, however, can be used with any other linear precoding in which the precoding matrix for each user is dependent only on the other users’ channels) [37] The MET approach enables the number of data streams for each user to be adaptively selected and at the same time avoids the complexity of joint iterative precoder/equalizer design [38] Denote K as the set of served users at a given time interval and assign indices k = 1, , |K | Denote Lk as the set of eigenmodes selected for transmission to user k, which are indexed from to k , where k = |Lk | Under a linear precoding framework, the total number of data streams in the downlink, denoted as the system transmission mode (STM), is upper bounded by the number of transmit antennas and can be written as |K | L= j, (4) j =1 where ≤ L ≤ BNt The aggregate transmitted signal is given by |K | x= Tk dk , k=1 (5) EURASIP Journal on Advances in Signal Processing where Tk ∈ CBNt × k is the precoding matrix and dk denotes the k dimensional signal vector for user k It is assumed that each user k at a given time slot is able to perfectly estimate its channel matrix Hk without any error Furthermore, each user k performs a singular value decomposition (SVD) on its channel as Hk = Uk Σk Vk We denote the ith singular value and the corresponding left and right singular vectors of Hk as σk,i , uk,i , and vk,i , respectively We also assume that the singular values in Σk are arranged in the descending order; that is, σk,1 ≥ σk,2 ≥ · · · ≥ σk,Nr The kth user’s receiver is a linear equalizer given by [Uk ]∗ k ) Using (3) and (5), the :(1: postprocessed signal rk after applying the linear equalizer is given by Definition Let Z denote a q × p complex matrix with q ≤ p and a common covariance matrix C = E{z j z∗ } for all j, j where z j is the jth column vector of Z The elements of two columns zi and z j are assumed to be mutually independent If the elements of Z are identically distributed as CN (0, 1) such that E{Z} = 0, then the Hermitian matrix ZZ∗ is a central Wishart matrix with p degrees of freedom and covariance matrix C, denoted as ZZ∗ ∼ CWq (p, C) 3.1 Approximation of a Linear Combination of Wishart Matrices Let Ys ∼ CWq (ps , Cs ) for s = 1, 2, , S be mutually independent central Wishart matrices Consider a linear combination S |K | rk = Fk Tk dk + Fk T j d j + wk , (6) j =1, j = k / ∗ where Fk = [Uk ]:(1: k ) Hk and wk is the processed noise, which is still white since the equalizer is a unitary matrix In the case of perfect knowledge of F1 , , F|K | , denote the aggregate interference matrix as Hk = [F∗ · · · F∗−1 F∗ · · · F∗ | ]∗ |K k k+1 To suppress the interuser interference, the constraint F j Tk = must be satisfied for k = j [39] This requires that Tk lies / in the null space of Hk With this constraint satisfied, the second term on the right hand side of the equality in (6) becomes zero Denote the total number of interfering data streams for user k from the other (|K | − 1) selected users | as k = |jK1, j = k j As a result, there are only BNt − k = / spatial degrees of freedom available at the transmitter side to support spatial multiplexing for user k, and therefore, k ≤ min{Nr , BNt − k } In [40], it was shown that the precoding matrix Tk can be written as a cascade of two precoding matrices Bk and Dk , that is, Tk = Bk Dk , where the BNt ×(BNt − k ) matrix Bk removes the interuser interference Denote the SVD of Hk as Hk = Uk Σk V(1) k V(0) k ∗ , (7) V(0) k corresponds to the right singular vectors of Hk where associated with the null modes One natural choice is Bk = V(0) As a matter of fact, Fk Bk is the effective interuser k interference-free channel for user k The (BNt − k ) × k matrix Dk is used for parallelization Denote the SVD of the effective channel for user k as (1) Fk Bk = Uk Σk Vk (0) ∗ Vk , Y= (8) (1) where Vk denotes the right singular vectors of Fk Bk corresponding to the first k nonzero singular values The (1) optimum choice of Dk is then Dk = Vk [41] Mathematical Preliminaries In this section, we present some mathematical preliminaries from matrix variate distributions and random matrix theory which prove useful in the analysis to follow For more detailed discussions, the readers are referred to [42–44] αs Ys , αs > (9) s=1 The distribution of Y can be approximated by the distribution of another Wishart matrix as Y ∼ CWq ( p, C) [42, page 124], where p is the equivalent degrees of freedom given by ⎡ (2q) S s=1 αs ps Cs ⎢ det p=⎣ det S s=1 αs ps (Cs ⊗ Cs ) ⎤1/q2 ⎥ ⎦ , (10) and C denotes the equivalent covariance matrix written as S C= αs ps Cs p s=1 (11) Now, if p1 = · · · = pS = p and C1 = · · · = CS = C, then (10) can be rewritten as ⎡ ⎢ p p=⎣ p ⎤1/q 2q2 S det (C)(2q) ⎥ s=1 αs ⎦ q2 S det(C ⊗ C) s=1 αs (12) Using the determinant property of the Kronecker product [42, Chapter 3], that is, det(C ⊗ C) = det(C)2q , in (12), we can obtain ⎡ ⎢ p = p⎣ 2⎤ S s=1 αs ⎥ ⎦ S s=1 αs (13) By substituting (13) in (11), it then holds that C= S s=1 αs S s=1 αs C (14) Finally, recall from Definition that the condition q ≤ p should hold for the Wishart distribution CWq ( p, C) to be meaningful In the following theorem, the upper and lower bound for p are obtained Theorem Assume that αs ≥ for s = 1, , S and also that at least one of the αs ’s is nonzero If p is defined in (13), then p ≤ p ≤ Sp Furthermore, the upper bound equality happens when α1 = α2 = · · · = αS , while the lower bound equality holds when ∃!s : αs > (∃! means there exists one and only one) Proof See Appendix A EURASIP Journal on Advances in Signal Processing 3.2 Truncation of Random Unitary Matrices and Jacobi Ensemble Definition If X ∼ CWm (n1 , C) and Y ∼ CWm (n2 , C) are independent complex Wishart matrices, then J = X(X + Y)−1 is called a complex Jacobi matrix It is shown in [45, Proposition 4.1] that J has the same distribution as that of [U](q,p) [U]∗ , where U ∈ U(n, n) (q,p) with q = m, p = n1 , and n = n1 + n2 Therefore, the eigenvalues of J are the same as those of [U](q,p) [U]∗ The (q,p) distribution of the extreme eigenvalues of the complex Jacobi ensemble is derived in [44] Ergodic Rate Analysis In this section, we derive an approximation for the ergodic rate of each user k at different modes k To assist the analysis, we assume that the elements of Hk,b are distributed such that Hk,b H∗ ∼ CWNr (Nt , C) for b = 1, , B Let the precoding k,b matrix for user k be written as Tk = Tk,1 · · · Tk,B , (15) where Tk,b denote the precoding applied at the bth BS for user k, such that the transmitted signal from the bth BS can | be written as xb = |K1 Tk,b dk Assuming MET and the k= practical per-BS power constraint (PBPC) with STM equal to L, the ergodic rate of a user k with k data streams using (6) can be expressed as Rk ( k , L) = E max log det I k + Fk Tk Qk T∗ F∗ k k Qk , Lemma Assume k = L− k , where L is the STM and k is the mode of user k Furthermore, assume Bk ∈ U(BNt , (BNt − k )) is the matrix that projects the channel of user k onto the null space of other users and is independent of Fk Assume Fk and Bk have SVDs given by Fk = UFk ΣFk V∗k and Bk = UBk ΣBk V∗k , F B respectively It then holds that ⎡ ⎡ k BP U k Σk L :(1: k ) Σk ∗ ∗ :(1: k ) Uk ⎤ BP = E⎣ log + λi (Fk Bk ) ⎦, L i=1 (a) (18) where (a) follows using the matrix identity det(I + AB) = det(I + BA) Therefore, the ergodic rate of a user k at mode k depends on the distributions of λi (Fk Bk ) for all i In order to compute the distribution of λi (Fk Bk ) in the network MIMO case, we provide the following result k f(λi (Fk ),λmin ) (λ, λ )dλ dλ ∞ log(λ) f(λi (Fk ),λmin ) (λ, λ )dλ dλ ∞ log(λ ) f(λi (Fk ),λmin ) (λ, λ )dλ dλ k BP + L i=1 log k + i=1 (a) BP f(λi (Fk ),λmin ) (λ, λ )dλ dλ L i=1 k BP λλ L ∞ i=1 + log log i=1 (17) = ∞ k = ∗ Rk ( k , L) = E log det I k + (19) ⎤ k k = subject to where Qk = E[dk dk ] is the power allocation matrix for user k and P is the power constraint at each BS Since the total power constraint (TPC) over all the BSs is less restrictive, the performance under TPC is equal or better than that under PBPC It has also been shown that there is only a marginal rate loss of PBPC to TPC [13] Therefore, for simplicity and analytical tractability, we assume TPC and equal power allocation among all the L data streams in the downlink, that is, Qk = (BP/L)I k The ergodic rate in (16), can be written as [37] ∀i BP λi (Fk )λmin ⎦ Rk ( k , L) ≈ E⎣ log L i=1 (16) for b = 1, , B, , We denote λmin ([V∗k UBk ](i,(BNt − k )) ) with λmin hereafter F in the paper for the ease of notation Denote the joint probability density function (pdf) of λi (Fk ) and λmin as f(λi (Fk ),λmin ) (λ, λ ), and let f(λi (Hk )) (λ) and f(λmin ) (λ ) denote the marginal pdf of λi (Fk ) and λmin , respectively Using the result of Lemma in (18) and the approximation log(1 + x) ≈ log(x), we can get an approximation for the ergodic rate for user k as k ≤ P, (i,(BNt − k )) Proof See Appendix B + tr E Tk,b Qk T∗ k,b V∗k UBk F λi (Fk Bk ) ≥ λi (Fk )λmin ∞ log(λ) f(λi (Hk )) (λ)dλ i=1 log(λ ) f(λmin ) (λ )dλ , (20) where (a) follows from the fact that λi (Fk ) = λi (Hk ) for all i, which results from (6) The elements of the aggregate channel matrix Hk in (1) for any realization of the kth user location not follow an i.i.d complex Gaussian distribution in general, due to the different large-scale channel coefficients to different BSs Assuming Hk,m and Hk,n are mutually independent for all m = n, Hk H∗ = B=1 ρk,b Hk,b H∗ is a linear combination / b k k,b of central Wishart matrices, and according to the results in Section 3, its distribution can be approximated with that of another Wishart matrix Hk H∗ ∼ WNr (Nt,k , ρk C), where k Nt,k and ρk are obtained using (13) and (14) as (since EURASIP Journal on Advances in Signal Processing the dimensions of Hk must be integers, we use x + 0.5 to round x to the nearest integer) Nt,k ⎢ ⎢ ⎢ ⎢ = p⎣ ⎥ ⎥ ⎥ + 0.5⎥, ⎦ B b=1 ρk,b B b=1 ρk,b where Γ(a, b) ¸ b xa−1 e−x dx and γ(a, b) ¸ xa−1 e−x dx denote the upper and lower incomplete Gamma functions, (n)(m) is given by respectively, αi, j ∞ ⎧ ⎪i + j − ⎪ ⎪ ⎪ ⎨ (21) and ρk = ⎝ if i < n and j < m, (n)(m) αi, j = i+ j ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ b ⎞ if i ≥ n and j ≥ m, i+ j−1 (26) otherwise, and B b=1 ρk,b ⎠ B b=1 ρk,b (22) b−1 ζa,b = a− j −1/a−b (27) j =0 Remark We note that Nt,k is a function of ρk,b for b = 1, , B, which depends on the position of the user k Therefore, for user k at any given position in the cell, the Nr × Nt,k i.i.d channel matrix Hk ∼ CN (0, ρk C) can be interpreted as if the user is communicating with one super BS with Nt,k transmit antennas and the equivalent large-scale channel coefficient ρk Furthermore, according to Theorem 1, the maximum of Nt,k is BNt , which corresponds to positions where ρk,1 = · · · = ρk,B At other positions, however, where user k experiences larger ρk,b values from some of the BSs and smaller from the others, Nt,k will be smaller than BNt It can be concluded that Nt,k is determined mainly by those BSs to which the user has largest ρk,b values, and those are the ones that help the cooperation and are actually seen by the user Since the distribution of Hk H∗ is approximated with k the distribution of another Wishart matrix Hk H∗ , we have k fλi (Hk ) (λ) ≈ fλi (Hk ) (λ) The distribution fλi (Hk ) (λ) for i = 1, , Nr for the uncorrelated central case is given in [46] as (the general framework developed in this paper, however, is applicable to arbitrarily correlated channels We only express the result for the uncorrelated case for simplicity) Nr fλucH ) (λ) = Guc ( i k Nr (−1)n+m λn+m−2+Nt,k −Nr e−λ |Ωuc |, n=1 m=1 (23) for i = 1, , Nr , where Guc is given by ⎡ Guc = ⎣ Nr Nr Nt,k − i ! i=1 ⎤−1 Nr − j !⎦ (24) j =1 The (i, j)th element of Ω is written as ωi, j ⎧ ⎪γ α(n)(m) + Nt,k − Nr + 1, λ ⎪ i, j ⎪ ⎪ ⎨ (n)(m) = Γ αi, j + Nt,k − Nr + 1, λ ⎪ ⎪ ⎪ ⎪ ⎩ (n)(m) αi, j + Nt,k − Nr !ζNr ,1 for i = 1, for i = Nr , otherwise, (25) In order to find fλmin (λ ), we note that the multiplication of two unitary matrices is another unitary matrix, that is, V∗k UBk ∈ U(BNt , BNt ) [47] Therefore, [V∗k UBk ](i,(BNt − k )) is F F a truncated unitary matrix As mentioned in Section 3.2, for any Wishart distributed unitary matrix with Haar measure A ∈ U(n, n), the multiplication [A](q,p) [A]∗ for q ≤ p ≤ (q,p) n, has the same distribution as a complex Jacobi ensemble [45, Proposition 4.1] The distribution of the minimum eigenvalues of the complex Jacobi ensemble is obtained in [44, Equation 3.2] fλmin (λ ) = Γi (BNt )Γi (i) Γi i + k Γi BNt − × i k λ (BNt − (1) ×2 F1 k −i−2) k (1 − λ )(i k −1) k − 1, i − BNt + k +2; k +i; (1 − λ )Ii−1 , (28) where Γm (c) = π m(m−1)/2 m Γ(c − j + 1) denotes the j= ∞ multivariate Gamma function, Γ(a) ¸ xa−1 e−x dx is the (1) Gamma function, and F1 ( k , i − BNt + k ; k + i; (1 − λ )Ii ) is a hypergeometric function of a matrix argument [44, 48] Based on (23) and (28), we can evaluate (20) numerically To verify the accuracy of the approximation in (20), we consider a hexagonal cellular layout with cell sectoring By using 120-degree sectoring in each cell, every neighboring cells can coordinate with each other to serve users in the shadow area shown in Figure The number of transmit antennas is chosen to be Nt = 4, which is the value currently implemented in wireless standards such as 3GPP LTE, and Nr = We randomly place two users in each cell sector The pathloss model is based on scenario C2 of the WINNER II specifications [49] The large-scale fading is modeled as lognormal with standard deviation of dB The edge SNR is defined to be the received SNR at the edge of the cell, assuming that one BS transmits at full power while all other BSs are off, accounting for pathloss but ignoring shadowing and small-scale fading Figure depicts the ergodic rate of a sample user k for k = 1, and different values of L (for the clarity of the plot we consider only some values of L The performance for the other values can be concluded easily) The results correspond EURASIP Journal on Advances in Signal Processing 12 Central unit 10 ^ Nt,k BS U2 U3 BS U1 0.2 BS 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized user distance from the home BS Figure 1: A network MIMO system with BSs, connected to a central unit via backhaul links Figure 3: Equivalent number of transmit antenna Nt,k versus the normalized distance from the home BS for a sample user k moving along the line that connects the BS to the center of the shaded hexagon in Figure 65 60 Sample user rate (bps/Hz) 55 50 45 40 L1 35 30 25 20 L2 15 −5 10 15 20 Edge SNR (dB) L = 1, approximation L = 1, simulation L = 5, approximation L = 5, simulation L = 10, approximation L = 10, simulation L = 2, approximation L = 2, simulation L = 6, approximation L = 6, simulation L = 11, approximation L = 11, simulation Figure 2: Ergodic rate of a sample user k versus edge SNR for a given realization of user locations, different values of L, and k = 1, The dashed lines indicate the rate obtained from the lower bound approximation, that is, Rk ( k , L), while the solid lines represent the rate obtained from the simulation, that is, Rk ( k , L) to one random snapshot of user locations when for any given L and k other users are assigned with or data streams It is also assumed that no user is within a normalized distance of 0.2 from its closest BS for the pathloss model to be valid It is observed that the lower bound approximation in (20) is very close to the simulation results obtained by Monte-Carlo simulations using (16) over the full range of edge SNR The difference between the approximation and the achieved rate is small enough to consider the approximation good enough for scheduling as explained in the next section To justify the argument in Remark 1, in Figure 3, we plot Nt,k versus the normalized distance from the home BS for a sample user k moving along the line that connects the BS to the center of the shaded hexagon in Figure It is observed that within a normalized distance of 0.5 from the BS 3, Nt,k = which results from the fact that ρk,3 is much larger than ρk,1 and ρk,2 , and therefore, only the BS is seen by this user As the user moves toward the center of the shaded hexagon, ρk,1 and ρk,2 increase but ρk,3 decreases, resulting an increase in Nt,k Indeed at the center of the shaded hexagon Nt,k = 12, which means all the BSs are seen by the user and actually can be helpful in the cooperation Therefore, BS cooperation is not very helpful for the cell interior user, and only edge users get most of the benefit This can be used to design BS cooperation Downlink Scheduling: Joint User and Mode Selection In this section, downlink scheduling for multimode transmission is discussed The total number of streams in the system under study is upper bounded by BNt , and normally BNt , so at each scheduling phase a subset of users, KNr and the preferred mode of each user must be selected for transmission In multimode transmission, the number of data streams for each user is adaptively selected, which allows to efficiently exploit the available degrees of freedom in the channel using multimode diversity (multimode diversity is a form of selection diversity among users with multiple antennas, which enables the scheduler to perform selection not only among the users (multiuser diversity), but also among the different eigenmodes of each user) [24] In a system with heterogenous users, the goal of the downlink scheduling is to make the system operate at a rate EURASIP Journal on Advances in Signal Processing point of its ergodic achievable rate region such that a suitable concave and increasing network utility function g(·) of the user individual ergodic rates is maximized [50] Let M = {1, 2, , KNr } denote the set consisting of all possible modes for all users, and let Si be a subset of M with |Si | ≤ BNt Let Ki denote the set of users with at least one selected mode in Si The downlink scheduling problem we wish to solve is defined as S = max g Si Rk ( k , |Si |) k = |Si |, 1≤ gMSRS Rk ( k , |Si |) |K | k∈Ki k ≤ Nr k=1 To solve (29) through brute-force exhaustive search over Si , BNt KNr m=1 Cm combinations must be checked Furthermore, for each combination Si , the knowledge of {Rk ( k , |Si |)}k∈Ki is required, which is not easy to compute in general This is a BNt and very computationally complex problem if KNr difficult to implement 5.1 Low-Complexity User and Mode Selection To reduce the computational complexity and at the same time exploit the benefits of multimode transmission, we propose a lowcomplexity joint user and mode selection algorithm To simplify the computation of {Rk ( k , |Si |)}k∈Ki for any given Si , we propose to use the approximations obtained in (20) for Rk ( k , |Si |) instead of the exact fomula in (16) This enables the analytical computation of g({Rk ( k , |Si |)}k∈Ki ) for any given Si , with only the knowledge of the average SNR of each users to all the BSs, and avoids the complexity of precoding matrix computations Therefore, it not only reduces the computational complexity at the central unit but also removes the need for any instantaneous CSI feedback at the expense of sacrificing the small-scale fading multiuser diversity One way to reduce the complexity associated with exhaustive search is to treat (29) as a relaxed optimization problem, that is, to greedily select data streams which maximize the network utility function Toward this goal, we gradually increase L from to BNt For any given L, the approximate ergodic rate for the next unselected eigenmode of all users is computed using (20) The algorithm continues until either L = BNt or the network utility function starts to decrease Once the user and mode selection is done, only selected users need to feedback the singular values and the corresponding singular vectors of their selected eigenmodes for precoder design Therefore, the proposed scheduling algorithm is of low complexity and is suitable for application when delay is not a stringent constraint or when the feedback resources is limited The resulting algorithm is summarized in Algorithm 5.2 Network Utility Function We focus on two special cases of network utility function, namely, the ergodic sum rate and the sum log ergodic rate To perform maximum sum rate = i Rk ( k , |Si |) B k=1 (30) To introduce fairness by performing proportional fairness scheduling (PFS) for a given Si , the per-cell sum log ergodic rate utility function is defined as [50] , (29) |Ki | subject to k∈Ki scheduling (MSRS) for a given Si , the per-cell ergodic sum rate utility function is defined as gPFS Rk ( k , |Si |) |K | k∈Ki = i log Rk ( k , |Si |) B k=1 (31) Simulation Results and Key Observations In this section, the performance of the proposed user and mode-selection strategy is evaluated via Monte-Carlo simulations The assumptions for the cellular layout, pathloss, shadowing, and the number of antennas are given in Section We drop K1 = K2 = K3 = 10 users randomly according to a uniform distribution in each cell Inspired from [49], we follow a drop-based simulation In this approach, a drop corresponds to one realization for user locations, during which the large-scale fading parameters as well as velocity and direction of travel for users, are practically constant Therefore, each user can only undergo small-scale fading at each location Furthermore, large-scale fading parameters are realized independently from drop to drop This method does not take into account the time evolution of the channel The main advantage of it is the simplicity of the simulation We run 1000 drops for user locations At the beginning of each drop, all the users feedback their average SNR from all the cooperating BSs (in real systems, such update is not frequent and only occurs when users move around) and the set S is obtained using the Algorithm For each obtained S at each drop, 1000 realizations are simulated with independent small-scale channel states 6.1 Sum Rates for Different Systems Figure 4, compares the ergodic sum rate of the proposed strategy with both MSRS and PFS to that of single-user transmission (SUT) and opportunistic scheduling based on instantaneous CSI (OSICSI) In SUT, only one user with the best ergodic rate is selected and served at each scheduling interval For the detailed information about the OSICSI algorithm, see [37] It is shown in Figure that the approximate sum rate is quite close to the achieved one with MSRS It can also be observed that the achieved sum rate for PFS is very close to that of MSRS It is further shown that the proposed algorithm for both MSRS and PFS performs much better than SUT and achieves a large fraction of the sum rate of OSICSI over a practical range of edge SNR values For example, at an edge SNR of 10 dB, it achieves 80% and 68% of the sum rate of opportunistic scheduling with MSRS and PFS, respectively In Figure 5, the sum log ergodic rate versus edge SNR is plotted It is observed that the approximate and simulated curves are in good agreement EURASIP Journal on Advances in Signal Processing (1) Initialization: L = 1, S = ∅, g(∅) = (2) while L ≤ min(KNr , BNt ) (3) r(L) = 0, ν = 0, SL = ∅, KL = ∅, Rk (i, L) = 0, k = 0, uk = for k = 1, , K and i = 1, , Nr (4) while ν ≤ L (5) for k = to K (6) Compute sk ← g({R j ( j , L)} j ∈KL−1 ∪ {Rk ( k + 1, L)}) from (20) j =k / (7) end for (8) k max ← arg maxk sk , k max ← k max + (9) r(L) ← s(k max), uk max ← 1, ν ← ν + (10) end while (11) for k = to K (12) if uk = then / (13) SL = SL ∪ { k } and KL = KL ∪ {k} (14) end if (15) end for (16) if r(L) < r(L − 1) then (17) S = SL−1, break (18) end if (19) L← L+1 (20) end while Algorithm 1: Pseudocode for the proposed algorithm 12 Sum log ergodic rate (bps/Hz/cell) 13 45 Ergodic sum rate (bps/Hz/cell) 50 40 35 30 25 20 15 10 −5 10 Edge SNR (dB) SUT PFS simulation MSRS simulation 15 20 11 10 −5 10 15 20 Edge SNR (dB) MSRS approximation OSICSI PFS-DMS Figure 4: Comparison of average sum rate versus edge SNR for B = 3, K = 30, Nt = 4, and Nr = To compare the performance of MSRS and PFS, we have divided the distance between users and their home BS into bins of width 0.1 of the cell radius, and for each bin, the fraction of times over all the 1000 drops that a user in that bin has been scheduled is plotted in Figure It is observed that with PFS, the activity fraction of users at farther distances from the BS is higher as compared to that using MSRS On the other hand, the simulation results show that the PFS algorithm almost always chooses data streams for each served user, while in MSRS data streams are also selected about 50% of the time that stream is selected Approximation Simulation Figure 5: Comparison of sum log ergodic rate versus edge SNR for B = 3, K = 30, Nt = 4, and Nr = Therefore, in PFS, choosing the users without considering the problem of the mode selection (dominant-mode transmission) seems to be the relevant strategy, at least in the chosen scenario The sum rate of PFS with only dominant selection (PFS-DMS) is also plotted in Figure It can be seen that the sum rate of PFS-DMS matches very well with that of PFS with user and mode selection 6.2 Feedback Analysis In this section, we compare the amount of feedback required by the proposed algorithm with 10 EURASIP Journal on Advances in Signal Processing BNt , corresponding to their selected eigenmodes at each small-scale realization for precoding design If the number of realizations of small-scale channel states within each drop is assumed to be T (1000 in this paper), the AFL is given by 25 Activity fraction (%) 20 15 ⎡ |K | ⎤ BK ⎦ Fprop = + ES ⎣ k (1 + 2BNt ) T k=1 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized user distance from the home BS (32) Since set S changes from drop to drop, the AFL is obtained by averaging over many drops For the opportunistic scheduling based on instantaneous CSI, however, all K users should feed back their respective Nr × BNt complex channel matrix in all the T realizations in each drop Therefore, we have MSRS PFS Fopp = 2BKNt Nr Figure 6: Comparison of activity fraction in percentage for both MSRS and PFS for edge SNR of 10 dB AFL per cell 103 102 10 15 20 25 30 Number of users per cell (33) In Figure 7, the AFL per cell versus the number of users per cell is compared for the proposed scheduling algorithm and the opportunistic scheduling based on instantaneous CSI It is observed that for the proposed algorithm, the AFL increases very slowly with the number of users, since no matter how big the number of users is in the cell, the number of served users will be limited by the maximum number of transmit antennas The only overhead of increasing the number of users is the average SNR values that should be fed back at the beginning of each drop, which is negligible with respect to the amount fed back during a drop For the opportunistic scheduling, however, the AFL increases linearly with the number of users since it has to feed back the CSI for all the users at each realization It is observed that for K1 = K2 = K3 = 30, AFL is decreased by more than 93% which makes the proposed algorithm attractive in such kind of scenarios Opportunistic scheduling Proposed algorithm Figure 7: Comparison of AFL per cell for the proposed algorithm and the opportunistic scheduling based on instantaneous CSI with edge SNR of 10 dB that by opportunistic scheduling based on instantaneous CSI We define the average feedback load (AFL) as the average number of real coefficients that are fed back to the central unit during each drop normalized to the number of small-scale realizations within that drop For the proposed algorithm (the results are almost the same for both MSRS and PFS since the AFL depends on the total number of transmitted data stream, that is, STM, which is the same on average for both schemes We only plot the result for MSRS here), once the set S is obtained at the beginning of a drop, each selected user k ∈ K sends back k real-valued singular values and k complex-valued right singular vectors of size Conclusion In this paper, we propose an analytical framework to approximate ergodic rates of users with different modes in a network MIMO system, based only on the knowledge of received average SNR from all the cooperating BSs at each user, called incomplete CSI in this paper Based on the derived approximate ergodic rates, the problem of downlink scheduling with both MSRS and PFS is addressed The proposed scheduling algorithm significantly reduces the feedback amount and performs close to the opportunistic scheduling based on instantaneous CSI It is of particular interest for applications where there is a total feedback overhead constraint and/or when there is no stringent delay constraint It is also shown that by introducing fairness, the probability of selecting higher modes for each users decreases significantly, which results in dominant mode transmission (beamforming) to each user EURASIP Journal on Advances in Signal Processing 11 Now, let us define V = v1 (Bk ), v2 (Bk ), , vi (Bk ) , where v j (Bk ) denotes the jth column of VBk and a1 , a2 , , an is the span of a1 , a2 , , an We have Appendices A Proof of Theorem In order to prove the lower bound for p, we write ⎡ λi (Fk Bk ) ≥ 2⎤ S s=1 αs ⎥ ⎦ S α2 s=1 s ⎢ p = p⎣ ⎡ S s=1 αs = p⎣ S s=1 αs S s=1 αs (a) ≥ p (a) S S j =1, j = i i=1 / S α2 s=1 s + αi α j x ∈V x =1 ⎤ ⎦ (A.1) ⎝ = ⎞⎛ S α2 ⎠⎝ s S ⎝ ⎞2 βs ⎠ (c) ΣFk V∗k UBk F x ∈V x =1 = ≥ ⎛ (A.2) (e) ≥ λi (Fk ) x ∈V x =1 ⎞ S α2 ⎠ s (A.3) s=1 ΣFk x ∈V x =1 s=1 αs ⎠ ≤ S⎝ s=1 =1 F k U Bk x ∈V x =1 (d) ⎞ Now, setting β1 = · · · = βS = 1, (A.2) can be rewritten as S Fk UBk IBNt ×m x = s=1 ⎛ x ∈V x αs βs ⎠ ≤ ⎝ s=1 (b) = p, ⎛ ⎞2 S = Fk UBk ΣBk x where (a) follows from the fact that αs > for all s The equality for the lower bound holds only when S S i=1 j =1, j = i αi α j = 0, and this happens only when ∃!s : / αs > The proof for the upper bound is established using the Cauchy-Schwarz inequality For some βs > for all s, we have ⎛ Fk Bk x x∈V x =1 (f) :(1:m) x VFk :(1,i) ∗ :(1:i) (i,m) U Bk :(1:m) x (i,m) x The upper bound for p using (A.3) is now obtained as ⎡ p ≤ p⎣ S S s=1 αs S s=1 αs (B.6) ⎤ ⎦ = Sp (A.4) The equality condition for the Cauchy-Schwarz inequality is only satisfied when the vectors α = [α1 · · · αS ] and β = [β1 · · · βS ] are linearly dependent; that is, α = xβ for some scalar x Since in our case β = [1 · · · 1], it results that the equality holds only when α1 = · · · = αS This completes the proof of the Theorem Here, (a) follows from the fact that (i) x ∈ V is equivalent to x ∈ V for some x = V∗k x; and (ii) x = x (b) B follows from the fact that the singular values of a unitary matrix are all equal to (c) follows from the fact that UFk y = y , where y = ΣFk V∗k [UBk ]:(1:m) x (d) follows F from the fact that defining z = [UBk ]:(1:m) x , we have ΣFk V∗k z F ΣFk = + B Proof of Lemma First, notice that for values of i > k or i > BNt − k , λi (Fk ), or λi (Bk ) are defined as zero Hence, the argument of this lemma is obvious for these cases Now, we prove the argument for i ≤ k For notational convenience, we define m = BNt − k According to “max-min” half of the CourantFischer Theorem [47], for any subspace M ⊂ CBNt − k with dimention i, we have λi (Fk Bk ) = max M,dim(M)=i Fk Bk x x∈M x =1 2 V∗k UBk F V∗k UBk F = λi (Fk )λmin :(1:m) x (B.5) :(1:i) ΣFk VFk ∗ :(1:i) z 1/2 :(i+1:BNt ) VFk ∗ :(i+1:BNt ) z (B.7) (e) follows from the fact that with q = [V∗k UBk ](i,m) x , we F have [ΣFk ]:(1:i) q ≥ λi (Fk ) q Finally, (f) follows from the 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Advances in Signal Processing and in part by SSF within the Strategic Research Center Charmant [15] References [1] S Venkatesan, A Lozano, and R Valenzuela, ? ?Network MIMO: overcoming intercell interference... step The two-step multimode transmission strategy was also proposed in the single-cell MIMO downlink in [22], with single-antenna users and imperfect CSI at the BS The main contributions are as follows